We prove global existence of smooth solutions of the 3D loglog
energy-supercritical wave equation $\partial_{tt} u - \triangle u = -u^{5}
\log^{c} (log(10+u^{2})) $ with $0 < c < {8/225}$ and smooth initial data
$(u(0)=u_{0}, \partial_{t} u(0)=u_{1})$. First we control the $L_{t}^{4}
L_{x}^{12}$ norm of the solution on an arbitrary size time interval by an
expression depending on the energy and an \textit{a priori} upper bound of its
$L_{t}^{\infty} \tilde{H}^{2}(\mathbb{R}^{3})$ norm, with
$\tilde{H}^{2}(\mathbb{R}^{3}):=\dot{H}^{2}(\mathbb{R}^{3}) \cap
\dot{H}^{1}(\mathbb{R}^{3})$. The proof of this long time estimate relies upon
the use of some potential decay estimates \cite{bahger, shatstruwe} and a
modification of an argument in \cite{taolog}. Then we find an \textit{a
posteriori} upper bound of the $L_{t}^{\infty} \tilde{H}^{2}(\mathbb{R}^{3})$
norm of the solution by combining the long time estimate with an induction on
time of the Strichartz estimates.